Lesson 9.2: Simplify Square Roots

New Concept

Today, we'll master simplifying square roots using the Product and Quotient Properties. This means rewriting radicals like $\sqrt{50}$ into a simpler form, such as $5\sqrt{2}$, by extracting all perfect square factors.

What’s next

Next, we will break down the process with step-by-step examples and interactive practice cards to build your skills.

Property

$\sqrt{a}$ is considered simplified if $a$ has no perfect square factors.

Examples

• $\sqrt{31}$ is simplified because 31 has no perfect square factors.

• $\sqrt{32}$ is not simplified because $32 = 16 \cdot 2$, and $16$ is a perfect square. The simplified form is $4\sqrt{2}$.

Property

If $a, b$ are non-negative real numbers, then $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$.

Simplify a square root using the product property.

1. Find the largest perfect square factor of the radicand. Rewrite the radicand as a product using the perfect-square factor.
2. Use the product rule to rewrite the radical as the product of two radicals.
3. Simplify the square root of the perfect square.

Examples

• To simplify $\sqrt{48}$: Find the largest perfect square factor, which is 16. Rewrite as $\sqrt{16 \cdot 3}$, which becomes $\sqrt{16} \cdot \sqrt{3}$, simplifying to $4\sqrt{3}$.

Property

An integer and a radical are not like terms and cannot be added together.
Similarly, radicals with different radicands cannot be added.
The expression $a + \sqrt{b}$ is fully simplified unless $\sqrt{b}$ can be simplified further.

Examples

• To simplify $5 + \sqrt{18}$: First, simplify the radical $\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}$. The expression is $5 + 3\sqrt{2}$, which is the final answer.

• To simplify $\sqrt{25} + \sqrt{81}$: Both are perfect squares, so simplify them first: $5 + 9 = 14$.

Property

A perfect square fraction is a fraction in which both the numerator and the denominator are perfect squares.

Examples

• To simplify $\sqrt{\frac{9}{64}}$: Since 9 and 64 are both perfect squares, we can find $\frac{\sqrt{9}}{\sqrt{64}}$, which is $\frac{3}{8}$.

• To simplify $\sqrt{\frac{121}{16}}$: Both 121 and 16 are perfect squares. The result is $\frac{\sqrt{121}}{\sqrt{16}} = \frac{11}{4}$.

Property

If $a, b$ are non-negative real numbers and $b \neq 0$, then

Simplify a square root using the quotient property.

1. Simplify the fraction in the radicand, if possible.
2. Use the Quotient Property to rewrite the radical as the quotient of two radicals.
3. Simplify the radicals in the numerator and the denominator.

Examples

• To simplify $\sqrt{\frac{75}{12}}$: First, simplify the fraction $\frac{75}{12} = \frac{25}{4}$. Now, $\sqrt{\frac{25}{4}} = \frac{\sqrt{25}}{\sqrt{4}} = \frac{5}{2}$.

• To simplify $\sqrt{\frac{21}{100}}$: The fraction cannot be simplified. Use the property to get $\frac{\sqrt{21}}{\sqrt{100}}$, which simplifies to $\frac{\sqrt{21}}{10}$.